Surjectivity of Gaussian maps for curves on Enriques surfaces

Abstract

Abstract. Making suitable generalizations of known results we prove some general facts about Gaussian maps. The above are then used, in the second part of the article, to give a set of conditions that insure the surjectivity of Gaussian maps for curves on Enriques surfaces. To do this we also solve a problem of independent interest: a tetragonal curve of genus g ≥ 7 lying on an Enriques surface and general in its linear system, cannot be, in its canonical embedding, a quadric section of a surface of degree g − 1 in P g−1 .

Brill–Noether theory of curves on Enriques surfaces I: the positive cone and gonality

“…We have M 2 −h M +2β+α−2 = 0 in all the cases in (26), and since h 0 (2M −L +K S ) ≤ α, we must have by (19) that α = h 0 (2M − L + K S ), h 0 (L − 2M) = 0 and therefore that equality occurs in (19), whence it must also occur in (25), so that…”

“…Since π 2,M is a dominant morphism, there must be a component J 0 of J M such that dim J 0 ≥ dim |L|, whence (19) follows. Assume now that equality occurs in (18).…”

“…They will be used to prove some cases of Theorem 1.3, to complete a theorem about plane curves in [18] and they will also be needed for the study of Gaussian maps in [19].…”

On the extendability of elliptic surfaces of rank two and higher

“…Let H S ∼ aC + bf be the hyperplane bundle of S and let g(S) be the sectional genus of S. We have: (3,7,13), (3,8,16), (3,10,22), (3,11,25), (3,13,31), (3,14,34), (4,9,21), (4,11,29), (4,13,37), (5,11,31), (5,12,36) (3,7,13), (3,8,16), (3,9,19), (3,10,22), (3,11,25), (3,12,28), (3,…”

“…To show (ii) we will exclude from the list in (iii) the five cases (a, b, g(S)) ∈ {(3, 9, 19), (3,12,28), (3,15,37), (4,10,25), (4, 12, 33)}.…”